Method for beam formation by calculation, in particular adapted to the compensation of failures of active modules of a radar with electronic scanning

ABSTRACT

The invention relates to a method for beam formation by calculation. For each defective active module of rank ip, the missing samples of the microwave signal a(îp) are calculated by one or more non-adaptive interpolations using the samples coming from the active modules in nominal operating mode situated in the neighborhood of the defective active modules, the beam being formed as if the interpolated samples a(îp) were the real measurements. In particular, the invention is applicable to the compensation for the effects of failures of one or more active modules distributed over an antenna of a radar with electronic scanning. The method according to the invention can notably be implemented within an airborne weather radar.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present Application is based on International Application No.PCT/EP2006/062621, filed May 24, 2006 which in turn corresponds toFrance Application No. 05 05211, filed on May 24, 2005, and priority ishereby claimed under 35 USC §119 based on these applications. Each ofthese applications are hereby incorporated by reference in theirentirety into the present application.

TECHNICAL FIELD

The invention relates to a method for beam formation by calculation. Inparticular, the invention is applicable to the compensation for theeffects of failures of one or more active modules distributed over anantenna of a radar with electronic scanning. The method according to theinvention can notably be implemented within an airborne weather radar.

BACKGROUND OF THE INVENTION

An antenna with electronic scanning can comprise a large number ofactive modules. Accordingly, in order to optimize the availability of aradar comprising an antenna with electronic scanning, the impact of thefailure of one or more active modules on the main functions of the radarmust be limited. It is thus desirable that the loss of several activemodules does not compromise the receiving function of the radar in orderto reach an optimum level of service. These constraints are justifiednotably when such a radar is used for applications requiring a highlevel of security and reliability of operation such as that required,for example, in the case of an airborne radar on a commercial aircraft,for example of the weather radar type.

In the case of a radar whose beam is formed by calculation, the full setof samples coming from the active modules is used in reception. When anactive module is defective due to a malfunction or fault, the samplescan no longer be employed for the formation of the beam withoutsignificantly degrading the reception performance of the radar. Thetolerance to these failures of active modules can notably be improved byusing interpolation methods for the spatial samples missing due tofailures. The radar beam is then formed by calculation using theinterpolations as if they were real samples.

For this purpose, there exist linear prediction methods which, by meansof the valid samples coming from the active modules in nominaloperation, allow the complete signal as it is received by the radar tobe decomposed into a sum of sinusoidal signals with amplitudes andfrequencies that said methods seek to estimate. Aside from the discreteFourrier transform, which does not directly provide this decomposition,the other linear interpolation techniques require the estimation ofcovariance matrices. These adaptive techniques may be readily applied toantennas whose active modules are uniformly distributed over the surfaceof the antenna.

However, the estimation of covariance matrices is complex and impreciseon an antenna whose active modules are distributed according to anon-constant distribution law over the surface of the antenna. Thelinear prediction methods are therefore maladapted to this type ofantenna due to their complexity and their cost.

SUMMARY OF THE INVENTION

The invention aims notably to overcome the aforementioned drawbacks. Forthis purpose, the subject of the invention is a method for formation bycalculation of a beam whose main lobe of a microwave signal is orientedin a direction Uzf pointed to by a scanning antenna comprising activemodules, the scanning being effected in one or more planes. The activemodules are identified by a rank i and by coordinates in a referencebase forming a plane within which the active modules of the antenna aresubstantially included. For each defective active module of rank ip, themissing samples of the microwave signal a(îp) are calculated by one ormore non-adaptive interpolations using the samples coming from theactive modules in nominal operating mode situated in the neighborhood ofthe defective active modules. The beam is formed as if the interpolatedsamples a(îp) were the real measurements.

In another embodiment, the active modules of a scanning antenna in aplane are arranged in rows and have as coordinates a position along anaxis perpendicular to the rows of active modules of the antenna. Theactive modules deliver, after sampling, samples a(i). The samples of themissing microwave signal a(îp) are defined according to the followingformula:

$\hat{a({ip})} = {{\frac{{z\left( {{{\mathbb{i}}\; p} + 1} \right)} - {z\left( {{\mathbb{i}}\; p} \right)}}{{z\left( {{{\mathbb{i}}\; p} + 1} \right)} - {z\left( {{{\mathbb{i}}\; p} - 1} \right)}} \times {a\left( {{\mathbb{i}p} - 1} \right)}} + {\frac{{z\left( {{\mathbb{i}}\; p} \right)} - {z\left( {{{\mathbb{i}}\; p} - 1} \right)}}{{z\left( {{{\mathbb{i}}\; p} + 1} \right)} - {z\left( {{{\mathbb{i}}\; p} - 1} \right)}} \times {{a\left( {{{\mathbb{i}}\; p} + 1} \right)}.}}}$The non-adaptive interpolation step can, for example, comprise thefollowing steps:

-   -   elimination of the phase gradient in the neighborhood of a        particular direction Uz1 of the samples of the microwave signal        coming from the active modules in nominal operating mode        situated in the neighborhood of the defective active modules;    -   non-adaptive linear interpolation of the missing samples of the        microwave signal;    -   rephasing of the samples obtained in the preceding step.

In another embodiment, the active modules of a scanning antenna in aplane are arranged in rows and have as coordinates a position (z(i))along an axis perpendicular to the rows of active modules of theantenna. The active modules deliver, after sampling, samples a(i). Thesamples of the missing microwave signal a(îp) are defined, for aparticular direction Uz1, according to the following formula:

$\hat{a({ip})} = {\left( {{\frac{{z\left( {{{\mathbb{i}}\; p} + 1} \right)} - {z\left( {{\mathbb{i}}\; p} \right)}}{{z\left( {{{\mathbb{i}}\; p} + 1} \right)} - {z\left( {{{\mathbb{i}}\; p} - 1} \right)}} \times {a\left( {{{\mathbb{i}}\; p} - 1} \right)}} + {\frac{{z\left( {{\mathbb{i}}\; p} \right)} - {z\left( {{{\mathbb{i}}\; p} - 1} \right)}}{{z\left( {{{\mathbb{i}}\; p} + 1} \right)} - {z\left( {{{\mathbb{i}}\; p} - 1} \right)}} \times {a\left( {{{\mathbb{i}}\; p} + 1} \right)}}} \right) \times {\exp\left( {{+ {j2\pi}}\;\frac{{Uz} - {{Uz}\; 1}}{\lambda}{z\left( {{\mathbb{i}}\; p} \right)}} \right)}}$

The method according to the invention can notably comprise the followingsteps:

-   -   formation of the beam by excluding the missing samples of the        microwave signal a(îp);    -   calculation by a non-adaptive linear interpolation of the        estimations of the samples of the missing microwave signal        a₁(îp), . . . , a_(P)(îp) starting from P assumptions of        particular direction Uz1, Uz2, . . . , Uzp;    -   calculation of the signals S₁ . . . S_(P) according to the        formula S₁=S₀+W(ip)×a₁(îp). . . ,        S_(P)=S₀+W(ip)×a_(P){circumflex over (()}ip) where W(ip)        corresponds to the weighting coefficient of the sample from the        defective module ip;    -   normalization of the signals S₁ . . . S_(P) in such a manner        that all the beams thus formed have the same gain in the        directed orientation Uzf;    -   selection of the signal S₁ . . . S_(P) whose measured power is        the lowest amongst the full set of calculated signals.        The number of particular directions can then be a function of        the number of defective active modules.

The active modules can for example be distributed according to anon-constant distribution law over the surface of the antenna.

The method according to the invention may notably be applied to a radardesigned for the detection and localization of weather phenomena.

The invention presents notably the advantage that the additional cost incomputation power resulting from the invention is very low compared tothat required by a channel formation without compensation.

BRIEF DESCRIPTION OF THE DRAWINGS

Other features and advantages of the invention will become apparent withthe aid of the description that follows with regard to the appendeddrawings which show:

FIG. 1, a distribution of rows of active modules of an antenna withelectronic scanning in one plane;

FIG. 2, a diagram representing the gain in elevation of the receivedsignal when the radar is in nominal operation;

FIG. 3, a diagram representing the gain in elevation of the receivedsignal when the radar is in impaired operation without correction;

FIG. 4, the steps of the method according to the invention;

FIG. 5, a diagram representing the gain in elevation of the receivedsignal and of the signal corrected by a method according to theinvention implementing a single linear interpolation when the radar isin impaired operation;

FIG. 6, the steps of the method according to the invention implementinga single linear interpolation optimized in the neighborhood of a givendirection;

FIG. 7, a diagram representing the gain in elevation of the receivedsignal and of the signal corrected by a method according to theinvention implementing a single linear interpolation optimized in theneighborhood of a given direction when the radar is in impairedoperation;

FIG. 8, the steps of the method according to the invention implementinga multiple linear interpolation;

FIG. 9, a diagram representing the gain in elevation of the receivedsignal and of the signal corrected by a method according to theinvention implementing a multiple linear interpolation when the radar isin impaired operation.

DETAILED DESCRIPTION OF THE DRAWINGS

FIG. 1 shows the distribution of the rows of active modules of anantenna with electronic scanning in one plane. The description thatfollows illustrates the method according to the invention taking as asupport for the description a radar equipped with an antenna withelectronic scanning in one plane. The method according to the inventionis more generally applicable to any radar comprising an antenna withactive modules.

FIG. 1 shows an orthogonal reference frame comprising a point O situatedin the same plane as the antenna at the center of the latter, togetherwith the axes OX, OY and OZ forming an orthogonal reference base, OY andOZ being in the plane of the antenna. All the coordinates mentioned inthe following description will be in this reference frame.

The antenna with electronic scanning illustrated in FIG. 1 comprises aset of rows 1, each row 1 comprising elementary sources 2. The rows 1comprise, in this example, a number N of elementary sources 2. Theposition on the axis OZ of any given row 1 of rank i, i being in therange between 0 and N−1, is denoted z(i). The phase center of all therows 1 is aligned with the axis OZ and, for a given row 1 of rank i, hasthe coordinates

${M(i)} = {\begin{bmatrix}0 \\0 \\{z({\mathbb{i}})}\end{bmatrix}.}$The distance between two rows 1 that are consecutive along the axis OZis not necessarily constant nor regular.

The operation described in the following corresponds to the case of acalibrated radar, in other words whose response in phase and inamplitude of all the elements is the same or only depends on thedifference in step due to the distribution of the active modules 2.Moreover, the individual diagram, denoted g({right arrow over (U)}), ofthe active modules 2 is identical.

When the radar receives a signal of wavelength λ and of power flux Φcoming from a direction

$\overset{\rightarrow}{U} = \begin{pmatrix}\sqrt{1 - {Uy}^{2} - {Uz}^{2}} \\{Uy} \\{Uz}\end{pmatrix}$identified by its direction cosines (Uy, Yz), a voltage a(i) is createdby this signal at the output of each row 1 of rank i. The voltage a(i)is proportional to

${\sqrt{\Phi} \times {g\left( \overset{\rightarrow}{U} \right)} \times {\exp\left( {{- \frac{2\pi}{\lambda}}{\overset{\rightarrow}{U} \cdot \overset{\rightarrow}{M({\mathbb{i}})}}} \right)}}\;$${or}\mspace{14mu}\sqrt{\Phi} \times {g\left( \overset{\rightarrow}{U} \right)} \times {{\exp\left( {{- \frac{2\pi \times {Uz}}{\lambda}}{z({\mathbb{i}})}} \right)}.}$A weighting denoted W(i) is assigned to each row 1 of rank i. Thedirection of formation of the main lobe oriented toward the targeteddirection is denoted Uzf. When the operation of the radar is nominal, inother words there are no defective active modules, the signal receivedby the radar corresponding to the contribution of all of the rows 1 isequal to:

$S = {\sum\limits_{i = 0}^{N - 1}{{a({\mathbb{i}})} \times {W({\mathbb{i}})} \times {\exp\left( {\frac{2\pi \times {Uzf}}{\lambda}{z({\mathbb{i}})}} \right)}}}$or$S \propto {\sqrt{\Phi} \times {g\left( \overset{\rightarrow}{U} \right)} \times {\sum\limits_{i = 0}^{N - 1}{{W({\mathbb{i}})} \times {{\exp\left( {\frac{2\pi \times \left( {{Uzf} - {Uz}} \right)}{\lambda}{z({\mathbb{i}})}} \right)}.}}}}$

FIG. 2 is a diagram representing the gain in elevation of the signalreceived when the radar is in nominal operation. The diagram comprisesan abscissa axis 10 indicating the direction cosine W relating to theangular difference with respect to the targeted direction Uzf. Theabscissa axis 10 is scaled from −1 to 1. The diagram comprises anordinate axis 11 indicating the relative power gain in elevation of thesignal received by the radar expressed in decibels per unit of power,the reference power, i.e. 0 dB/p, being assigned to the gain received inthe targeted direction Uzf. The ordinate axis 11 is scaled from 0 to −60dB/p. In this diagram, a curve C_(R) represents the gain in elevation ofthe received signal for a given direction cosine, the radar being innominal operation. Beam formation by calculation tends to form a mainlobe oriented in the targeted direction while at the same timeminimizing the gain in the other directions potentially carrying sourcesof interference.

FIG. 3 is a diagram representing the gain in elevation of the receivedsignal when the radar is in impaired mode and no correction is applied.The diagram comprises an abscissa axis 10 indicating the directioncosine W relating to the angular difference with respect to the targeteddirection Uzf. The abscissa axis 10 is scaled from −1 to 1. The diagramcomprises an ordinate axis 11 indicating the relative power gain inelevation of the signal received by the radar expressed in decibels perunit of power, the reference power, i.e. 0 dB/p, being assigned to thegain received in the targeted direction Uzf. The ordinate axis 11 isscaled from 0 to −60 dB/p. In this diagram, a curve C_(R2) representsthe gain in elevation of the received signal for a given directioncosine, the radar being in impaired mode and no correction beingapplied. The diagram illustrates the degradations suffered by an antennacomprising 35 rows 1, two of which are for example defective. Thecomparison between the diagram in FIG. 2 and the diagram in FIG. 3 showsan increase between the nominal mode and the impaired mode, illustratedby a rise in the secondary lobes 31 of around 30 dB above the gain ofthe interference noise signals arriving angular-shifted by 0.3 indirection cosine.

FIG. 4 shows an example of a series of steps resulting in theimplementation of the method according to the invention. In a samplingstep 100, the microwave signals coming from the active modules 2 aresampled. The samples obtained at step 100 coming from the active modulesin a nominal operating state are subsequently completed in step 101 withsamples of interpolated microwave signals corresponding to the defectiveactive modules 2. All of the samples available following steps 100 and101 are subsequently used by a step 102 in order to form the beam bycalculation in the direction Uzf targeted by the radar.

The method according to the invention additionally comprises the step101 for linear interpolation of missing spatial samples. The linearityof the interpolation step 101 according to the invention allows otherspectral analysis methods to be used downstream. The failure of at leastone active module 2 of a row 1 leads to the loss of one sample a(îp). Anestimation of this sample a(îp) can be calculated using the samples fromits neighborhood by a linear operation. Thus, it can be written that:

$\hat{a({ip})} = {f\left\lbrack \;{\ldots\mspace{11mu},{a\left( {{{\mathbb{i}}\; p} - 2} \right)},{a\left( {{{\mathbb{i}}\; p} - 1} \right)},{{{a\left( {{{{\mathbb{i}}\; p} + 1},{a\left( {{{\mathbb{i}}\; p} + 2} \right)},\ldots}\mspace{11mu} \right\rbrack}{or}\hat{a({ip})}} = {\sum\limits_{{k = {- P}},{k \neq {ip}}}^{+ P}{{\alpha(k)} \cdot {{a\left( {{{\mathbb{i}}\; p} + k} \right)}.}}}}} \right.}$

In the case where there is only one defective signal source, the seriesof received signals a(i) corresponds to the sampling, possiblyirregular, of a spatial sinusoid according to the points z(i). Thespatial frequency of this sinusoid is given by the term Uz/λ. To withinthe noise which is assumed to be independent from one source to another,the samples a(i) are disposed on a circle in the complex plane.

According to the method according to the invention, the missing samplesare estimated by linear interpolation in a non-adaptive fashion, inother words where their interpolation does not depend on the variationof the received samples over time. The modulus of the radar signal isdenoted A whereas the thermal noise relating to one row 1 of rank i isdenoted b(i). One step of the method according to the invention amountsto recovering the missing sample

${a\left( {{\mathbb{i}}\; p} \right)} = {A\;{\exp\left( {{- {j2\pi}}\;\frac{Uz}{\lambda}{z\left( {{\mathbb{i}}\; p} \right)}} \right)}}$with the knowledge of the samples adjacent to the defective one equal to

${a\left( {{{\mathbb{i}}\; p} - 1} \right)} = {{A\;{\exp\left( {{- {j2\pi}}\;\frac{Uz}{\lambda}{z\left( {{{\mathbb{i}}\; p} - 1} \right)}} \right)}} + {b\left( {{{\mathbb{i}}\; p} - 1} \right)}}$and${a\left( {{{\mathbb{i}}\; p} + 1} \right)} = {{A\;{\exp\left( {{- {j2\pi}}\;\frac{Uz}{\lambda}{z\left( {{{\mathbb{i}}\; p} + 1} \right)}} \right)}} + {{b\left( {{{\mathbb{i}}\; p} + 1} \right)}.}}$The missing sample can notably be calculated in step 101 by a singlelinear interpolation of the method according to the invention with thefollowing formula

$\hat{a({ip})} = {{\frac{{z\left( {{{\mathbb{i}}\; p} + 1} \right)} - {z\left( {{\mathbb{i}}\; p} \right)}}{{z\left( {{{\mathbb{i}}\; p} + 1} \right)} - {z\left( {{{\mathbb{i}}\; p} - 1} \right)}} \times {a\left( {{{\mathbb{i}}\; p} - 1} \right)}} + {\frac{{z\left( {{\mathbb{i}}\; p} \right)} - {z\left( {{{\mathbb{i}}\; p} - 1} \right)}}{{z\left( {{{\mathbb{i}}\; p} + 1} \right)} - {z\left( {{{\mathbb{i}}\; p} - 1} \right)}} \times {{a\left( {{\mathbb{i}p} + 1} \right)}.}}}$In step 102, the beam is formed as if the interpolated sample a(îp) werethe real measurement. The interpolation error ε is then substantiallyequal to

${\cos\left\lbrack {\pi \times \frac{{z\left( {{{\mathbb{i}}\; p} + 1} \right)} - {z\left( {{{\mathbb{i}}\; p} - 1} \right)}}{\lambda}{Uz}} \right\rbrack} - 1$if the distribution of the active modules 2 is substantially regular.

FIG. 5 is a diagram representing the gain in elevation of the receivedsignal and of the signal corrected by a method according to theinvention implementing a single linear interpolation such as isdescribed hereinabove when the radar is in impaired operation. Thediagram comprises an abscissa axis 10 indicating the direction cosine Wrelating to the angular difference with respect to the targeteddirection Uzf. The abscissa axis 10 is scaled from −1 to 1. The diagramcomprises an ordinate axis 11 indicating the relative power gain inelevation of the signal received by the radar expressed in decibels perunit of power, the reference power, i.e. 0 dB/p, being assigned to thegain received in the targeted direction Uzf. The ordinate axis 11 isscaled from 0 to −60 dB/p. In this diagram, a curve C_(R3) representsthe gain in elevation of the received signal for a given directioncosine, the radar being in nominal operation. In this diagram, a curveC_(Corr) represents the gain in elevation of the received signalcorrected by a method according to the invention implementing a singlelinear interpolation such as was previously described for a givendirection cosine, the radar being affected by the failure of onedefective source. As can be observed in FIG. 4, this embodiment proveseffective in the case where the direction of arrival of the signals Uzis close to 0.

The preceding description illustrates an example where only one row 1 ofactive modules 2 is defective. The same embodiment of the methodaccording to the invention can be used to compensate for the failure ofseveral rows 1 of active modules 2. In such a case, the same processingoperation as described hereinabove is applied to each row 1 of defectiveactive modules 2. Moreover, if several consecutive rows 1 of activemodules 2 are defective, the rows 1 in nominal operation surrounding thedefective rows are used for the interpolation. In the case of a radarcomprising an antenna with electronic scanning in several planes, theinterpolation is effected on all of the adjacent elements in thedirection of the axis OZ, but also in the direction of the axis OY.

FIG. 6 shows the steps of the method according to the inventionimplementing a single linear interpolation optimized in the neighborhoodof a given direction. The interpolation steps 101 are adapted in orderto enhance the efficiency of the rejection of the noise signals in agiven direction Uz1. Step 101 can notably comprise three steps. In astep 110, the method according to the invention eliminates the phasegradient in the neighborhood of a particular direction Uz1 of thesamples of the microwave signal coming from the active modules 2 innominal operating mode situated in the neighborhood of the defectiveactive modules 2 then, in a step 111, calculates a new interpolatedvalue. This interpolated value is subsequently rephased in a step 112 asa function of the position z(ip) of the faulty element. The missingsample a₁(îp) may be calculated from the following equation:

$\hat{a({ip})} = {\left( {{\frac{{z\left( {{{\mathbb{i}}\; p} + 1} \right)} - {z\left( {{\mathbb{i}}\; p} \right)}}{{z\left( {{{\mathbb{i}}\; p} + 1} \right)} - {z\left( {{{\mathbb{i}}\; p} - 1} \right)}} \times {a\left( {{{\mathbb{i}}\; p} - 1} \right)}} + {\frac{{z\left( {{\mathbb{i}}\; p} \right)} - {z\left( {{{\mathbb{i}}\; p} - 1} \right)}}{{z\left( {{{\mathbb{i}}\; p} + 1} \right)} - {z\left( {{{\mathbb{i}}\; p} - 1} \right)}} \times {a\left( {{{\mathbb{i}}\; p} + 1} \right)}}} \right) \times {\exp\left( {{+ {j2\pi}}\;\frac{{Uz} - {{Uz}\; 1}}{\lambda}{z\left( {{\mathbb{i}}\; p} \right)}} \right)}}$The sampling and the formation of the beam by calculation in step 102 iscarried out in an identical manner to the embodiment previouslydescribed.

FIG. 7 is a diagram representing the gain in elevation of the receivedsignal and of the signal corrected by a method according to theinvention implementing a single linear interpolation such as isdescribed hereinabove optimized in the neighborhood of a given directionwhen the radar is in impaired operation. The diagram comprises anabscissa axis 10 indicating the direction cosine W relating to theangular difference with respect to the targeted direction Uzf. Theabscissa axis 10 is scaled from −1 to 1. The diagram comprises anordinate axis 11 indicating the relative power gain in elevation of thesignal received by the radar expressed in decibels per unit of power,the reference power, i.e. 0 dB/p, being assigned to the gain received inthe targeted direction Uzf. The ordinate axis 11 is scaled from 0 to −60dB/p. In this diagram, the curve C_(R4) represents the gain in elevationof the received signal for a given direction cosine, the radar being innominal operation. In this diagram, the curve C_(Corr2) represents thegain in elevation of the received signal for a given direction cosinecorrected by a method according to the invention implementing a singlelinear interpolation optimized in the neighborhood of a given direction,the radar being affected by the failure of one faulty source. FIG. 5illustrates a correction optimized in the direction Uz1 equal to −0.5radians.

The preceding description illustrates an example where a single row 1 ofactive modules 2 is defective. The same embodiment of the methodaccording to the invention can be used to compensate for the failure ofseveral rows 1 of active modules 2. In such a case, the same processingoperation as described hereinabove is applied to each row 1 of defectiveactive modules 2. Moreover, if several consecutive rows 1 of activemodules 2 are defective, the rows 1 in nominal operation surrounding thedefective rows are used for the interpolation. In the case of a radarcomprising an antenna with electronic scanning in several planes, theinterpolation is effected on all of the adjacent elements in thedirection of the axis OZ, but also in the direction of the axis OY inorder to enhance the efficiency of the rejection of the noise signals ina given solid angle oriented with respect to a direction Uz1.

FIG. 8 shows the steps of the method according to the inventionimplementing a multiple linear interpolation. The sampling step 100 isidentical to the other embodiments. Similarly, the interpolation step101 comprising notably the step 110 for elimination of the phasegradient in a given direction, the single interpolation step 111 and therephasing step 112 are identical to the preceding embodiment. The methodaccording to the invention in the example in FIG. 8 carries out a numberP of interpolations of the samples corresponding to the defective rowsbased on P assumptions of particular directions Uz1, Uz2, . . . , Uzp byrepeating step 101 described previously. For each missing sample of rankip, a number P of estimations a₁(îp), . . . , a_(P)(îp) are obtained. Atstep 102, the beam is subsequently formed by excluding the faultysamples using the formula

$S_{0} = {\sum\limits_{i \neq {ip}}{{W({\mathbb{i}})} \times {{\exp\left( {\frac{2\pi \times \left( {{Uzf} - {Uz}} \right)}{\lambda}{z({\mathbb{i}})}} \right)}.}}}$The beam is completed with the P interpolations of the faulty activemodules 2, which yields the following P results: S₁=S₀+W(ip)×a₁(îp), . .. , S_(P)=S₀+W(ip)×a_(P)(îp). The resulting signals S₁ . . . S_(P) aresubsequently normalized at step 120 in such a manner that all the beamsthus formed have the same gain in the directed orientation Uzf, in otherwords in the direction pointed to by the antenna. From the resultingsignals S₁ . . . S_(P), the signal whose power calculated in step 121 isthe lowest amongst all of the calculated signals is retained in step122, in other words the signal j, j being in the range between 1 and P,corresponding to the equation |S_(J)|=min(|S₁|, |S₂|, . . . , |S_(P)|).

As the gain from all the diagrams is identical in the main lobe, thecriterion amounts to minimizing the gain in the direction of theinterference. This minimization is valid for several interference noisesignals as long as they are relatively close to one another. In the caseof a waveform with no ambiguity in distance, for example of the LowFrequency of Recurrence (LFR) type, in airborne applications, thisassumption is practically always verified.

The formation of the beam by calculation in step 102 is carried out inan identical manner to the embodiment previously presented. In thisembodiment, the greater the number of defective rows 1 of active modules2, the more advantageous it is to increase the number of assumptions ofparticular directions Uz1, Uz2, . . . , Uzp.

FIG. 9 is a diagram representing the gain in elevation of the receivedsignal and of the signal corrected by a method according to theinvention implementing a multiple linear interpolation such as isdescribed hereinabove optimized in the neighborhood of a given directionwhen the radar is in impaired operation. The diagram comprises anabscissa axis 10 indicating the direction cosine W relating to theangular difference with respect to the targeted direction Uzf. Theabscissa axis 10 is scaled from −1 to 1. The diagram comprises anordinate axis 11 indicating the relative power gain in elevation of thesignal received by the radar expressed in decibels per unit of power,the reference power, i.e. 0 dB/p, being assigned to the gain received inthe targeted direction Uzf. The ordinate axis 11 is scaled from 0 to −60dB/p. In this diagram, the curve C_(R5) represents the gain in elevationof the received signal for a given direction cosine, the radar being innominal operation. In this diagram, the curve C_(Corr3) represents thegain in elevation of the received signal for a given direction cosinecorrected by a method according to the invention implementing a multiplelinear interpolation, the radar being affected by the failure of onedefective source.

The preceding description illustrates an example where a single row 1 ofactive modules 2 is defective. The same embodiment of the methodaccording to the invention can be used to compensate for the failure ofseveral rows 1 of active modules 2. In such a case, the same processingoperation as described hereinabove is applied to each row 1 of defectiveactive modules 2. Moreover, if several consecutive rows 1 of activemodules 2 are defective, the rows 1 in nominal operation surrounding thedefective rows are used for the interpolation. In the case of a radarcomprising an antenna with electronic scanning in several planes, theinterpolation is effected on all of the adjacent elements in thedirection of the axis OZ, but also in the direction of the axis OY inorder to enhance the efficiency of the rejection of the noise signals inone of the given solid angles oriented with respect to the directionsUz1, Uz2, . . . , Uzp.

A radar implementing an embodiment of the method according to theinvention can notably be airborne. In addition, the method according tothe invention can, for example, be used in the steps for processing thesignal received by a weather radar. The method according to theinvention can notably be implemented by a digital computer.

It will be readily seen by one of ordinary skill in the art thatembodiments according to the present invention fulfill many of theadvantages set forth above. After reading the foregoing specification,one of ordinary skill will be able to affect various changes,substitutions of equivalents and various other aspects of the inventionas broadly disclosed herein. It is therefore intended that theprotection granted hereon be limited only by the definition contained inthe appended claims and equivalents thereof.

The invention claimed is:
 1. A method of determining a radar beam basedupon a scanning antenna having active modules wherein at least one ofthe active modules is defective, the method comprising: orienting a mainlobe of a microwave signal in a direction Uzf pointed to by a scanningantenna comprising active modules, the scanning being effected in one ormore planes; sampling and identifying the active modules by a rank i andby coordinates in a reference base forming a plane within which theactive modules of the antenna are substantially included, wherein theactive modules are distributed according to a non-constant distributionlaw over a surface of the antenna; determining for each defective activemodule of rank ip, missing samples of the microwave signal a(îp)relating to the defective active module, by calculating one or morenon-adaptive interpolations using the samples coming from the activemodules in nominal operating mode situated in the neighborhood of thedefective active modules, the beam being formed as if the interpolatedsamples a(îp) were the missing samples; wherein the non-adaptiveinterpolation step comprises the steps of: reducing the phase gradientequal to zero in the neighborhood of a particular direction Uz1 of thesamples of the microwave signal coming from the active modules innominal operating mode situated in the neighborhood of the defectiveactive modules; non-adaptive linear interpolation of the missing samplesof the microwave signal; and rephasing of the missing samples obtainedin the preceding step.
 2. The method as claimed in claim 1, wherein themethod comprises the following steps: forming the beam by excluding themissing samples of the microwave signal a(îp); calculating using anon-adaptive linear interpolation of the estimations of the samples ofthe missing microwave signal a₁(îp),. . ., a_(P)(îp) starting from Passumptions of particular direction Uz1, Uz2, ..., Uzp; calculatingsignals S₁ . . . S_(P) according to the formula S₁=S₀+W(ip)×a₁(îp), . .., S_(P)=S₀+W(ip)×a_(P)(îp) where W(ip) corresponds to the weightingcoefficient of the sample from the defective module ip; normalizing thesignals S₁ . . . S_(P) in such a manner that all the beams thus formedhave the same gain in the directed orientation Uzf; selecting the signalS₁ . . . S_(P) whose measured power is the lowest amongst the full setof calculated signals.
 3. The method as claimed in claim 2, wherein themethod of claim 2 is applied to a radar designed for the detection andlocalization of weather phenomena.
 4. A method of determining a radarbeam based upon a scanning antenna having active modules wherein atleast one of the active modules is defective, the method comprising:orienting a main lobe of a microwave signal in a direction Uzf pointedto by a scanning antenna comprising active modules, the scanning beingeffected in one or more planes; sampling and identifying the activemodules by a rank i and by coordinates in a reference base forming aplane within which the active modules of the antenna are substantiallyincluded, wherein the active modules are distributed according to anon-constant distribution law over a surface of the antenna; determiningfor each defective active module of rank ip, missing samples of themicrowave signal a(îp) relating to the defective active module, bycalculating one or more non-adaptive interpolations using the samplescoming from the active modules in nominal operating mode situated in theneighborhood of the defective active modules, the beam being formed asif the interpolated samples a(îp) were the missing samples; wherein theactive modules of a scanning antenna in a plane are arranged in rows andhave as coordinates a position along an axis perpendicular to the rowsof active modules of the antenna, the method includes the active modulesdelivering, after sampling, samples a(i), the samples of the missingmicrowave signal a(îp) are defined, for a particular direction Uz1,according to the following formula:$\hat{a({ip})} = {\left( {{\frac{{z\left( {{{\mathbb{i}}\; p} + 1} \right)} - {z\left( {{\mathbb{i}}\; p} \right)}}{{z\left( {{{\mathbb{i}}\; p} + 1} \right)} - {z\left( {{{\mathbb{i}}\; p} - 1} \right)}} \times {a\left( {{{\mathbb{i}}\; p} - 1} \right)}} + {\frac{{z\left( {{\mathbb{i}}\; p} \right)} - {z\left( {{{\mathbb{i}}\; p} - 1} \right)}}{{z\left( {{{\mathbb{i}}\; p} + 1} \right)} - {z\left( {{{\mathbb{i}}\; p} - 1} \right)}} \times {a\left( {{{\mathbb{i}}\; p} + 1} \right)}}} \right) \times {\exp\left( {{+ {j2\pi}}\;\frac{{Uz} - {{Uz}\; 1}}{\lambda}{z\left( {{\mathbb{i}}\; p} \right)}} \right)}}$being the rank of a defective active module.
 5. The method of claim 4,comprising the following steps: forming the beam by excluding themissing samples of the microwave signal a(îp); calculating, by anon-adaptive linear interpolation of the estimations of the samples ofthe missing microwave signal a₁(îp),. . .,a_(P)(îp) starting from Passumptions of particular direction Uz1, Uz2, . . ., Uzp; calculatingsignals S₁ . . . S_(P) according to the formula S₁=S₀+W(ip)×a₁(îp), . .., S_(P)=S₀+W(ip)×a_(P)(îp) where W(ip) corresponds to the weightingcoefficient of the sample from the defective module ip; normalizingsignals S₁ . . . S_(P) in such a manner that all the beams thus formedhave the same gain in the directed orientation Uzf; selecting the signalS₁ . . . S_(P) whose measured power is the lowest amongst the full setof calculated signals.
 6. The method as claimed in claim 5, wherein anumber of particular directions of the radar signal is a function of thenumber of defective active modules.
 7. The method as claimed in claim 5,wherein it is applied to a radar designed for the detection andlocalization of weather phenomena.
 8. A method of determining a radarbeam based upon a scanning antenna having active modules wherein atleast one of the active modules is defective, the method comprising:orienting a main lobe of a microwave signal in a direction Uzf pointedto by a scanning antenna comprising active modules, the scanning beingeffected in one or more planes; sampling and identifying the activemodules by a rank i and by coordinates in a reference base forming aplane within which the active modules of the antenna are substantiallyincluded, wherein the active modules are distributed according to anon-constant distribution law over a surface of the antenna; determiningfor each defective active module of rank ip, missing samples of themicrowave signal a(îp) relating to the defective active module, bycalculating one or more non-adaptive interpolations using the samplescoming from the active modules in nominal operating mode situated in theneighborhood of the defective active modules, the beam being formed asif the interpolated samples a(îp) were the missing samples; arrangingthe active modules of a scanning antenna in a plane area in rows andhaving as coordinates, a position along an axis perpendicular to therows of active modules of the antenna; the active modules delivering,after sampling, samples a(i), the samples of the missing microwavesignals a(îp) are defined according to the following formula:$\hat{a({ip})} = {{\frac{{z\left( {{{\mathbb{i}}\; p} + 1} \right)} - {z\left( {{\mathbb{i}}\; p} \right)}}{{z\left( {{{\mathbb{i}}\; p} + 1} \right)} - {z\left( {{{\mathbb{i}}\; p} - 1} \right)}} \times {a\left( {{{\mathbb{i}}\; p} - 1} \right)}} + {\frac{{z\left( {{\mathbb{i}}\; p} \right)} - {z\left( {{{\mathbb{i}}\; p} - 1} \right)}}{{z\left( {{{\mathbb{i}}\; p} + 1} \right)} - {z\left( {{{\mathbb{i}}\; p} - 1} \right)}} \times {{a\left( {{{\mathbb{i}}\; p} + 1} \right)}.}}}$wherein ip is the rank of a defective active module.
 9. A method ofdetermining a radar beam based upon a scanning antenna having activemodules wherein at least one of the active modules is defective, themethod comprising: orienting a main lobe of a microwave signal in adirection Uzf pointed to by a scanning antenna comprising activemodules, the scanning being effected in one or more planes; sampling andidentifying the active modules by a rank i and by coordinates in areference base forming a plane within which the active modules of theantenna are substantially included; determining for each defectiveactive module of rank ip, missing samples of the microwave signal a(îp)relating to the defective active module, by calculating one or morenon-adaptive interpolations using the samples coming from the activemodules in nominal operating mode situated in the neighborhood of thedefective active modules, the beam being formed as if the interpolatedsamples a(îp) were the real measurements and, wherein the active modulesof a scanning antenna in a plane area are arranged in rows and have ascoordinates a position along an axis perpendicular to the rows of activemodules of the antenna, said active modules delivering, after sampling,samples a(i), the samples of the missing microwave signals a(îp) aredefined according to the following formula:${a\left( {\hat{i}p} \right)} = {{\frac{{z\left( {{ip} + 1} \right)} - {z({ip})}}{{z\left( {{ip} + 1} \right)} - {z\left( {{ip} - 1} \right)}} \times {a\left( {{ip} - 1} \right)}} + {\frac{{z({ip})} - {z\left( {{ip} - 1} \right)}}{{z\left( {{ip} + 1} \right)} - {z\left( {{ip} - 1} \right)}} \times {a\left( {{ip} + 1} \right)}}}$ip being the rank of a defective active module.